Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-8x+4y &= 4 \\ -2x-3y &= -6\end{align*}$
We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $3$ and the bottom equation by $4$ $\begin{align*}-24x+12y &= 12\\ -8x-12y &= -24\end{align*}$ Add the top and bottom equations. $-32x = -12$ Divide both sides by $-32$ and reduce as necessary. $x = \dfrac{3}{8}$ Substitute $\dfrac{3}{8}$ for $x$ in the top equation. $-8( \dfrac{3}{8})+4y = 4$ $-3+4y = 4$ $4y = 7$ $y = \dfrac{7}{4}$ The solution is $\enspace x = \dfrac{3}{8}, \enspace y = \dfrac{7}{4}$.